The function must have an average value of zero. Mathematically, this is expressed as:

\[\int_{-\infty}^{\infty} \psi(t) \, dt = 0\]

This condition ensures that the wavelet can detect changes or “details” in the signal rather than its average or constant components.

The function \(\psi(t)\) must be square integrable, meaning it has finite energy:

\[\int_{-\infty}^{\infty} |\psi(t)|^2 \, dt < \infty\]

This requirement ensures that the wavelet’s energy is finite, making it possible to localize the function in both time and frequency domains. Functions that satisfy this belong to the \(L^2(\rm I\!R)\) space, which is the space of all functions with finite energy.

The wavelet’s Fourier transform, \(\hat{\psi}(\omega)\), should satisfy the admissibility condition:

\[C_\psi = \int_{-\infty}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{|\omega|} \, d\omega < \infty\]

where \(\hat{\psi}(\omega)\) is the Fourier transform of \(\psi(t)\), and \(\omega\) represents angular frequency. This condition implies that \(\hat{\psi}(\omega)\) must approach zero as \(\omega \rightarrow 0\) meaning the wavelet has no component at zero frequency (or DC component). This condition is crucial for ensuring that the wavelet transform is invertible.

A mother wavelet should generally be oscillatory or “wavelike” (hence the term “wavelet”). This oscillatory behavior allows the wavelet to capture variations in the signal. For example, wavelets like the Morlet wavelet resemble decaying sinusoids. This oscillatory nature helps the wavelet capture both high-frequency and low-frequency components effectively.

Although not strictly necessary, compact support is often a desirable property. Compact support means that the function is non-zero only over a finite interval, making it well-localized in time. This allows for efficient computation and good localization in the time domain. For example, the Haar wavelet has compact support, while others, like the Morlet wavelet, do not have strict compact support but still decay rapidly.